Rounding decimals

Classroom resources

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Students will be able to

• I can round a decimal to a number of decimal places.
• I can say which digit decides, and why halfway rounds up.
• I can choose how much to round for the job, and explain my choice.

Timing

Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Students need decimal place value (the place value lesson and Year 6).

The one idea: rounding picks the nearer of two numbers with the right number of places, and the digit straight after the place shows which. The common mistakes are cutting digits off (3.74), letting later digits push the deciding digit up (Kai's double rounding), and counting places wrongly (rounding to 1 place when 2 was asked).

Node 2 is judgement: rounding to the cent for money, up for tins of paint, to the ruler's accuracy for measurements. Accept any justified answer in class discussion.

Running the lesson

1. Predict: students who chose 3.74 cut the digits off; 3.7 is one place too few.
2. Run: slide the decimal places down and ask what the two ends of the line are each time.
3. Investigate: ask what Kai's 2.35 and 2.4 would mean if 2.349 were a real length.
4. Modify: for the paint, ask what happens if she buys 3 tins.
5. Make: collect the smallest and largest numbers that round to 4.57.

The hard part (on purpose)

Deciding how much to round in context. Ask 'what is this number for, and how accurately could we have measured it?' before any digits are cut.

When students get stuck

Answers to the checks

1. round 3.746: Round 3.746 to 2 decimal places.
   3.75. The digit after the hundredths is 6, so the 4 goes up to 5. 3.74 cuts the digits off.
2. where does it sit?: Set the decimal places to 2. Is 3.746 before or past the halfway mark? What does it round to?
   
   3.746 is past the halfway mark 3.745, so it is nearer 3.75: it rounds to 3.75.
3. Round it: Round 8.372 to 1 decimal place.
   8.4
   The digit after the tenths is 7, so round up: 8.4.
4. Find the mistake: Kai rounds 2.349 to 1 decimal place. Click the line where the working goes wrong.
   line 2
   Line 2: only the digit straight after the tenths matters, and that is 4, so round down. Kai let the 9 push the 4 up, then rounded again. 2.349 to 1 decimal place is 2.3.
5. Halfway: Round 6.45 to 1 decimal place.
   6.5
   The next digit is 5, exactly halfway, and halfway rounds up: 6.5.
6. The stopwatch: A stopwatch gives 12.846 seconds. Round it to the nearest tenth of a second.
   
   12.846 to 1 decimal place is 12.8: the next digit is 4.
7. Enough paint: A painter needs 14.2 m² of coverage. One tin covers 4 m², so 14.2 ÷ 4 = 3.55 tins. How many tins should she buy?
   4
   Here you round up to the next whole tin: 4, so that there is enough.
8. Dollars and cents: A shop's total is $7.499. What is charged, to the nearest cent?
   $7.50
   Cents are 2 decimal places. The digit after is 9, so 7.49 goes up to 7.50.
9. The ruler: A ruler marked in millimetres measures a pencil. The calculator gives 14.2857 cm. Which is the most sensible way to report it?
   14.3 cm, to the nearest millimetre
   One millimetre is 0.1 cm, so the ruler supports 1 decimal place: 14.3 cm.
10. Three places: Round 5.0763 to 3 decimal places.
    5.076
    The digit after the thousandths is 3, so 5.076.
11. The 9 does not count: Round 7.0449 to 2 decimal places.
    7.04
    The next digit is 4, so 7.04.
12. Share the bill: Three friends share a $50 bill: $50 ÷ 3 = 16.666… Each pays this much, to the nearest cent. (Write it as 16.xx.)
    16.67
    Cents are 2 decimal places. The next digit is 6, so 16.67.
13. Review: powers of 10: Review: write $3 \times 10^2 + 5 \times 10^0$ as a number.
    305
    300 + 5 = 305. The tens column is 0.
14. lands on 4.57: Set the number so that, to 2 decimal places, it rounds to 4.57.
    
    Any number from 4.565 to 4.5749… rounds to 4.57, for example 4.568 or 4.571.

Solutions

One possible solution for each task. Other correct programs are fine.

how much is enough? › The stopwatch

Open task: any program that does what the step asks.

how much is enough? › Enough paint

Open task: any program that does what the step asks.

how much is enough? › Dollars and cents

Open task: any program that does what the step asks.

how much is enough? › The ruler

Open task: any program that does what the step asks.

lands on 4.57

Open task: any program that does what the step asks.

Stretch and support

Stretch: Round at every step of 3.46 + 3.46 to the nearest whole number, then round only at the end. Do you get the same answer? Which should you trust?

Support: Keep the number line in view and say which two numbers it is between before choosing.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand the ideas in ONE lesson only, from the lesson "Rounding decimals".

The ideas:
- Rounding decimals to specified places: I can Students round decimals to a given number of decimal places using standard rounding conventions.
- Context-appropriate accuracy: I can Students choose an appropriate level of rounding for a given context and explain why that accuracy is suitable.

What understanding looks like:
- Correctly rounds a decimal to a specified number of decimal places.
- Applies the 'round half up' convention consistently.
- Does not confuse rounding to 2 decimal places with rounding to the nearest 100.
- Selects a contextually appropriate level of decimal accuracy and justifies it.
- Recognises that different contexts require different precision.
- Can explain why rounding to too many places can give a false sense of precision.

Mistakes students often make here:
- A student cuts digits off (3.74).
- A student rounds 2.349 to 2.35 and then to 2.4 (Kai's slip).
- A student reports a ruler measurement to 4 decimal places.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Rounding decimals".

Exit ticket

A stick is measured as 83.46 cm with a ruler marked in millimetres. Round it sensibly, and say why.

83.5 cm: the ruler shows millimetres, which is 1 decimal place of a centimetre.

What each step is for

• round 3.746: Committing to an answer shows whether they cut the digits off (3.74), look at the right digit (3.75) or round the wrong place (3.7).
• where does it sit?: Seeing 3.746 on a line between 3.74 and 3.75, past the halfway mark, shows why the next digit decides: 5 or more is past halfway.
• one digit decides: Only the digit straight after the place matters; Kai rounds the 9 into the 4 and then again. Halfway rounds up.
• how much is enough?: The right amount of rounding depends on the job: a timer, a purchase and a ruler each suggest a different place, and sometimes 'enough' means always round up.
• rounding: Rounding to 3 places, to the cent in a share, and rounding to 2 places where the next digit is small; the review keeps place value fresh.
• lands on 4.57: Finding a number that rounds to 4.57 shows a whole range of numbers, 4.565 up to just under 4.575, round to the same value.

Where it leads

Next: Estimating, using rounded numbers to check an answer.

Where it leads: significant figures and accuracy in Year 8 and 9, and error in measurement.